Karen Gillan and Arthur Darvill are squealing like happy toddlers, eagerly rummaging through big carrier bags of CDs on the floor of the Guardian Guide's office. "Hey, The Hit World Of Marthas And Arthurs!" says Arthur, waving a sleeve.

"Definitely have that one," laughs Karen. They both chew gum as they pore over piles of discs, Arthur next picking out a Richard Hawley single. "Who is Richard Hawley?" Karen asks, blowing at her new fringe. Arthur gawps incredulously. When I mention that these CDs are going spare, the noise levels go through the roof. With their time as Amy and Rory coming to an end, the soon-to-be-ex Doctor's assistants have been this hyper since they arrived. They burst into our photographer's studio half-an-hour late, emitting a first-coffee-of-the-morning buzz. She is all eyes wide and long limbs crossing and un-crossing. He is polite, low-key and more guarded, with occasional bursts of noise. It's impossible not to stare at Karen. She is taller than Big Ben. She dabs on extra powder before the shoot and Arthur joins her at the mirror to give his quiff a final waxing. She nobly ditches her platform brogues to even out the height difference between them. As the shoot goes on, his velvet jacket gets gradually covered in the fuzz from her fluffy jumper. She, meanwhile, is the kind of woman who can wear mohair with a leather skirt and opaque tights on a thick summer's day and not sweat at all. As soon as the photographer starts clicking, the pair swing into action. Arthur tells Karen he just did a photoshoot for a prominent gay magazine and ended up wearing a skirt. "I can't wait to see the photos!" Karen yelps, clapping her hands.

It was April 2010 when the Doctor first crash-landed his Tardis in Amelia Pond's back garden but it seems like longer ago, probably because so much has been packed into the two-and-a-half years since (and also because time has no meaning in Steven Moffat's infuriatingly fluid universe). Amy (her grown-up name) is mysteriously without parents, utterly self-sufficient and ready for adventure. She is also impossibly foxy and totally unafraid. Rory, meanwhile, has his work cut out to prove himself more than a timid fiance who lives in the shadow of her obsession with the Doctor. They will soon be leaving in "heartbreaking circumstances" after one final encounter with evil, time-sucking effigies the Weeping Angels. When they finally go, both say they'll be gone for good.

Since the show's reboot in 2005, the Doctor has had three other permanent companions, played by Billie Piper, Freema Agyeman and Catherine Tate. But none have quite caught the public attention like Amy and Rory. Amy's not in love with the Doctor, unlike Rose (Piper) or Martha (Agyeman), which means she can get on with the important business of running down spaceship corridors in Doc Martens, casting glib one-liners over her shoulder. Rory's not just along for the ride, either.

Amy did make one attempt at seducing the Doctor, but Karen insists their relationship was "more a deep connection with someone from childhood. Someone that you look up to so much and have obsessed about, so it's more than just fancying him."

Arthur says this actually made it harder for Rory, because the other man wasn't just a bit of a looker. "How do you compete with that?" he asks. "But I think Rory did compete. And he won."

"Won?" Karen gasps. "It's not a competition."

There's smirking: "Yeah, just for the sake of argument, he did."

'It's like being in a band. You go through so much stuff together. I'd hate to do it on my own' – Karen Gillan

Photograph: Chris McAndrew for the Guardian

In 2010 it was Karen's casting as Amy that made the headlines. She was to be the feisty new assistant with all the attendant drooling over her model's looks. Rory was initially just the fiance she couldn't commit to, especially not when the Doctor came calling to whisk Amy into space. It was only later that he blossomed into a bumbling action hero with his own fair share of one-liners. Arthur's favourite episode is, he says, The Pandorica Opens because he gets to play a Roman centurion robot with a gun in his finger. "I felt like I was filming Indiana Jones," he enthuses. "Playing a soldier in space!"

But weren't you just a big space gooseberry, I ask him. "Space gooseberry!" they shout in unison. This happens a lot when they're talking. At one point in the interview they're just meowing at each other. Two years doing the time-warp does funny things to a person. When we first caught up with this trans-dimensional Mr & Mrs in the current series they were on the brink of divorce. In real life, however, Karen and Arthur are a hooting, karate-chopping double act who are clearly going to miss each other. I ask how any Doctor Who assistant could be happy going back to reality after living it up in the Tardis. "Yeah, how do you carry on with your life?" ponders Arthur. When I remind him he's about to find out, he lights up. "This will always be the biggest thing that ever happened to us."

"It's like being in a band," says Karen. "You go through so much stuff and you go to amazing places and you do it all together. I'd hate to do it on my own. It'd feel really lonely, actually." But she is leaving the band to go solo. "Oh yeah," she goofs. "I'm scared."

They may not have topped the charts, but the pair are treated like the Beatles when they go to conventions or bump into fans on the street. While at ComicCon in San Diego earlier this year, Matt Smith and the Ponds decided to crash a party run by the makers of the Doctor Who Tumblr page (it's the second-biggest after Beyoncé's, which delights Karen). "They didn't know it was going to happen," she reveals.

"We just turned up. I have never seen a reaction like it," adds Arthur. "We couldn't hear anything because they just screamed."

Karen's in full flight now, remembering the rush: "Oh my god, it was just like …" She makes a prolonged, high-pitched noise like a trapped horse. She is relentlessly energetic, hollering and whooping to illustrate her points. Arthur is more reserved and often mutters agreement good-naturedly, letting her take the limelight. It's not unlike their on-screen partnership.

Arthur Darvill did toy with a music career back in the day and formed a band called Edmund with his friends. He named it after his favourite character in The Chronicles Of Narnia and, judging by their presence on YouTube – an ardent Who fan has uploaded some of their songs – they sound like the Arctic Monkeys fronted by a baby James Dean Bradfield. Arthur was, naturally, the lead singer and guitarist. "Edmund's dead now. We've had a funeral," he says, smiling. It was always going to be acting for him. But music is something he comes back to in his down time.

The US in particular has embraced the weird man and his chums in the blue box with gusto since the arrival of Matt Smith, Karen and Arthur. The last ever scenes they filmed for the show were shot in New York and the locals turned out in force when they heard the Doctor was coming to Central Park. "It's weird over there," says Karen. "It's like this hip, underground thing that they like in Brooklyn and Williamsburg which is funny because it's not like that here."

"But the last time we went there something had changed," continues Karen. "People were coming up to us on the street the whole time saying they loved the show. There were paparazzi following us around which never happens over there. When we got to Central Park there were just hundreds and hundreds of people." Arthur nods, looking off into the distance, gently yeah-ing.

No one was expecting such interest and the Who crew hadn't bothered to arrange any security to keeps fans from mobbing the show's stars. "I just remember running to the car, like I was in a band or something," says Karen. "People were like [American accent] 'Sign my Tardis!' We just thought, 'This is never going to happen again so we should enjoy it.'"

They wrapped filming in New York a couple of months ago and dived straight into the pile of job offers awaiting them. Karen's already done a film, Not Another Happy Ending, and is about to start shooting the third instalment of Charlie Brooker's spoof detective show A Touch Of Cloth. Then it's off to America, to star in new supernatural flick Oculus. Arthur has just finished filming his role as a "geordie pervert" in a forthcoming BBC drama and is about to straddle theatre and TV simultaneously, doing a West End play while filming ITV1's Broadchurch with ex-Doctor David Tennant. They are striking while the iron is hot.

The role of Doctor's assistant is unlike any other acting job on British television. In the last year, two actors who've done tours of duty as Tardis sidekicks – Mary Tamm and Caroline John – have passed away; they were still called "former Doctor Who assistants" in every obituary and news story. I ask Karen and Arthur if they'll still be happy to be called "former Doctor Who assistants" when their careers have taken them far away from Cardiff and Steven Moffat. Will they still go to conventions?

"I wouldn't mind because I quite like conventions," says Karen. "I don't understand how people could complain about meeting all the people who enjoy what you do. They're so much fun. I met Brent Spiner who plays Data from Star Trek …"

Which all sounds like larks, but isn't it going to be annoying to be referred to as Amy and Rory when you're trying to break Hollywood or auditioning for the National Theatre? "Oh, yawn-fest," blurts Karen, taking Arthur by surprise. He erupts into snickers.

Armed with their modest haul of CDs, the pair disappear into the lift, designer shades on, and head out into the sunshine for a smoke before their separate cars arrive. Time to go solo.

Its stories might be getting darker, and its plots ever-more outlandish, but the Time Lord still has millions of fellow travellers. Helen Lewis-Hasteley and Andrew Harrison debate the tea-time institution